Tuesday, August 5, 2014

Edwin Moise on postulates considered as self-evident truths

My favorite book I've read this summer has hands-down been Edwin E. Moise's, Elementary Geometry From An Advanced Standpoint (yes, I know, I'm a nerd). I'm nowhere near finished with it, but in planning for teaching Geometry this year, I've used it a lot to help me think through what we assert and why we assert it.

This section is a beautiful summary of the many wonders and problems with an axiomatic system such as the one we continue to teach. I don't know exactly how I'm going to use this, but it is definitely going to inform my presentation of something.

pp. 282-3


In the time of Euclid, and for over two thousand years thereafter, the postulates of geometry were thought of as self-evident truths about physical space; and geometry was thought of as a kind of purely deductive physics. Starting with the truths that were self-evident, geometers considered that they were deducing other and more obscure truths without the possibility of error. (Here, of course, we are not counting the casual errors of individuals, which in mathematics are nearly always corrected rather promptly.) This conception of the enterprise in which geometers were engaged appeared to rest on firmer and firmer ground as the centuries wore on. As the other sciences developed, it became plain that in their earlier stages they had fallen into fundamental errors. Meanwhile the “self-evident truths” of geometry continued to look like truths, and also continued to seem self-evident.

With the development of hyperbolic geometry, however, this view became untenable. We then had two different, and mutually incompatible, systems of geometry. Each of them was mathematically self-consistent, and each of them was compatible with our observations of the physical world. From this point on, the whole discussion of the relation between geometry and physical space was carried on in quiet different terms. We now thing not of a unique, physically “true” geometry, but of a number of mathematical geometries, each of which may be a good or bad approximation of physical space, and each of which may be useful in various physical investigations. Thus we have lost our faith not only in the idea that simple and fundamental truths can be relied upon to be self-evident, but also in the idea that geometry is an aspect of physics.

This philosophical revolution is reflected, oddly enough, in the differences between the early passages of the Declaration of Independence and the Gettysburg Address. Thomas Jefferson wrote:

  “ . . . We hold these truths to be self-evident, that all men are created equal, that they are endowed by their creator with certain unalienable rights, that among these are Life, Liberty and the pursuit of Happiness . . . . “

The spirit of these remarks is Euclidean. From his postulates, Jefferson went on to deduce a nontrivial theorem, to the effect that the American colonies had the right to establish their independence by force of arms.

Lincoln spoke in a quite different style:

  “Fourscore and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition that all men are created equal.”

Here Lincoln is referring to one of the propositions mentioned by Jefferson, but he is not claiming, as Jefferson did, that this proposition is self-evidently true, or even that it is true at all. He refers to it merely as a proposition to which a certain nation was dedicated. Thus, to Lincoln, this proposition is a description of a certain aspect of the United States (and, of course, an aspect of himself). (I am indebted for this observation to Lipman Bers.)

This is not to say that Lincoln was a reader of Lobachevsky, Bolyai or Gauss, or that he was influenced, even at several removes, by people who were. It seems more likely that a shift in philosophy had been developing independently  of the mathematicians, and that this helped to give mathematicians the courage to undertake non-Euclidean investigations and publish the results.

At any rate, modern mathematicians use postulates in the spirit of Lincoln. The question where the postulates are “true” does not even arise. Sets of postulates are regarded merely as descriptions of mathematical structures. Their value consists in the fact that they are practical aids in the study of the mathematical structures that they describe.

Wednesday, July 30, 2014

"The organism moves towards health" — reflections on TMC14

Everybody is writing blog posts about feeling like a fraud after an amazing experience at Twitter Math Camp 2014. Impostor syndrome. I feel like a fraud too, at least, most of the time, but I am trying to practice refraining from my conditioned habits of reacting automatically and giving in in response to that defense mechanism. I am practicing not-reacting. I am trying to notice the positive energy that is there and to just allow it. I am trying to allow myself to experience myself as a competent, good-enough teacher I have respect for and want to continue to be.
me practicing accepting myself as a competent,
good-enough teacher, seen here
with supportive tweeps & a giant margarita

What worked in the Group Work Working Group session was setting up a structure to sustain that positive energy of presence. Having learned how to do that is a huge gift I have given myself over the past 25 years of dharma practice. It's a "gift" that comes from working very hard at being present and practicing every day, rain or shine, whether I feel like it or not, whether I do it well or do it badly. I follow the three teachings my teacher Natalie Goldberg learned from her teacher Katagiri Roshi: Continue under all circumstances. Don't be tossed away. Make positive effort for the good. I have done that in my practice every single day for over 25 years. It's the one thing I know in my feet that I am good at.
starting with restorative classroom circles

So I decided to bring THAT to Twitter Math Camp this year.

The structure works because it is a structure for teaching and sustaining presence — learning to be present with an open heart. I have dharma sisters and brothers all over the world, but when we practice together online, we practice asynchronously — each of us on our own, in our own lives, in our own homes. When we come together using the asynchronous forum of online communication, we maintain that same structure of presence. Write when you write. Read when you read. Listen when you listen.

No comment.

using the Talking Points structure; no comment
"No comment" is the most important part of the structure, and the hardest part to implement online in a forum like Twitter, which is designed to support comments. "No comment" is about allow there to be space for everyone. It is about all being present together authentically and about staying present with whatever arises. THAT is the thing that most adolescents don't get exposed to in their lives, and it is the thing that can make the greatest possible difference in the quality of their experience — both in the math classroom and everywhere else in their lives.

Most people in our culture don't have a lot of experience in being present and staying present. It takes an enormous amount of energy to learn how to stay present and not flinch. But do that with anything you love and you will have a magical experience. Do that with math, and you'll unlock the treasures of your amazing human mind. But being present with others in a big open space is hard. At first it can be scary. It's very naked. That is why the structure of "no comment" is so important. It helps to create a shared space of emotional safety. It gets everybody focused on their own stuff and supports dropping the "act" you bring to most in-person interactions. That's why it's good to do so right from the start. It's about reframing our conditioned habits of personality.

Very quickly, the timed structure and the practice of "no comment" makes the practice of presence very freeing. You begin to relax into that big open space. You become curious. Your defenses soften. You begin to notice the interesting patterns of your own mind. Best self and worst self. Curious self and bored self. Zen mind and monkey mind. Defense mechanisms, such as snark.

collaborative mathematics
using the Talking Points structure; no comment
The practice of "no comment" creates a space in which the authentic thoughts of your own amazing human mind can arise and step forward. And we honor that process by persisting in not-commenting as we continue.

Natalie describes this process as stepping forward with your own mind.

Once you get a taste for being present, you'll naturally begin to crave it more. That is something I count on in my classroom management practice. Fred always said, "The organism moves towards health." That is one of his greatest teachings for me. "The organism moves towards health" means that, in the process of growing up, we all fall away from the naturally sane and healthy patterns of our organism. "Fight or flight" is a falling away from the natural discharge cycle of "rest and digest" we experienced as infants. When you're hungry, you eat. When you're tired, you sleep. Fred said there is a deeper wisdom inside us that is always available for us to tap back into. It's like an underground stream that is part of our psychological and emotional water table. When we practice being present through structures like Talking Points or meditation or writing practice, it feels like a homecoming — a homecoming to a natural state that is healthy and inquisitive and curious to see what will happen next. It is a natural reconnection with our own inner growth mindset that is our birthright — not some artificial fantasy state we impose on students from without by telling them to have one.

assigning competence after group work & 
observation; still no comment
A growth mindset is just the psyche's way of attuning to the fundamental idea that our organism moves naturally in the direction of health if we will let it — if we can get out of its way and allow it to unfold as it needs to. Allowing means learning to refrain from interfering with that natural movement, and so we use structures that make it manageable for ordinary human beings like us to access the extraordinary ocean of intellectual and creative possibility that is mathematics.

Ten minutes at a time is about what I can muster, I have learned over the years.
Kate test-driving a geometry task using the
Talking Points structure; still no comment
In my experience teaching meditation and writing practice and other structures that cultivate presence, I have found it is about what most people can handle. Ten minutes of Talking Points, no comment — GO. Ten minutes of writing practice — keep your hand moving, no comment, GO. Ten minutes of mathematical conversation, no comment, GO. Learning how to be present with the big, scary openness of not-knowing is no small thing. That is why we zone out, check our phones a hundred times an hour, play video games, watch TV, assault-eat, numb out, zone out, distract ourselves. We all crave the real stuff, but connecting with it feels like sticking a butter knife into the electrical socket. So we break it into more manageable chunks. We set a limit for ourselves and dive in for a limited period. We practice being present for ten minutes at a time. And then we give ourselves and our students a break. It helps us build our tolerance for the intensity of presence and it builds our courage to come back and try it again the next time.

Natalie says that monkey mind is the guardian at the gate, protecting the treasures of our heart and strengthening us for the challenge of opening ourselves to presence and to Big Mind. The structure of "no comment" makes it feel safe for us to touch in to that fire at the center of our being. It helps us to close the gap between what we THINK we've been doing and what we have ACTUALLY done. For me, it's about strengthening students' courage to open their hearts to contact with their amazing mathematical minds — with what my friend Max Ray of The Math Forum at Drexel calls their "mathematical imaginations." We math teachers know the secret that everyone has this mathematical imagination. Our greatest challenge is to get students to trust that they have it too and can access it safely and reliably.


Read Taming Your Gremlin by Rick Carson.

Tuesday, July 22, 2014

TMC #14 GWWG – Annotated References: the research on explicit teaching of exploratory talk

OK, last summarizing post before I have to get packing.

Here is the background research on ‘exploratory talk. Once again, please note that this is NOT required reading!  Recreational reading only! So please don't freak out!  :)

I wanted to provide links and titles to valuable materials.

These are listed in order of relevance to the Group Work Working Group morning session — they are not in formal bibliographical form.

Shell Centre,  MAP PD: Students Working Collaboratively (PD Module 5)
An adequate introduction, with pointers to some of the major reseach on exploratory talk in the math classroom. PD module on using talk in the math classroom. There are two parts to download: the overview and the “handouts for teachers.” The Handouts for Teachers is more useful than the overview, but you kind of need both to get started.

Neil Mercer and Steve Hodgkinson, editors, Exploring Talk in School
The richest single source of research on cultivating exploratory talk, though many of the chapters I found most useful are available on the internet.

Douglas Barnes, “Exploratory Talk for Learning” (chapter 1 in Exploring Talk in School, but downloadable PDF here)
Barnes is the research who originally pioneered the concept of exploratory talk as distinguished from cumulative talk and disputational talk. Everybody else’s work builds on his.

Neil Mercer and Lyn Dawes, “The value of exploratory talk” (chapter 4 in Exploring Talk in School, but downloadable PDF here)
An explanation of the need for differing kinds of talk in the classroom (asymmetrical teacher-student talk as well as symmetric student-student or peer talk). Makes a strong case for doing the work to intentionally raise the level of symmetrical talk to make group work effective and meaningful.

Lyn Dawes, The Essential Speaking and Listening,  Chapter 2: “Talking Points” (downloadable PDF here). 
Overview of her original strategy for using the “Talking Points” activity structure to encourage students to develop what Barnes referred to as “an open and hypothetical style of learning.” This is what I used as the basis for developing my own version of the Talking Points activity.

Thinking Together
How can children be explicitly enabled to use talk more effectively as a tool for reasoning? The Thinking Together program was founded to address this specific question. A freely available, evidence-based interventional program of three talk lessons plus supporting materials, developed by University of Cambridge Faculty of Education researchers (led by Mercer and Dawes) as “a dialogue-based approach to the development of children’s thinking and learning.” It has been widely implemented in the U.K. as a means of cultivating exploratory talk in the classroom. Their materials are targeted at elementary children, but can be adapted for older students as well.

Sylvia Rojas-Drummond and Neil Mercer, “Scaffolding the development of effective collaboration and learning”
Summary of a collaboration between Mexican and British researchers that documents the effectiveness of using exploratory talk strategies to improve collaborative reasoning and learning in the group work-centered classroom.

Neil Mercer and Claire Sams, “Teaching children how to use language to solve maths problems”
Summary of the research behind the interventional program called Thinking Together  and how to proactively make talk-based group activities more effective in developing students’ mathematical reasoning, understanding, and problem-solving.

Monday, July 21, 2014

#TMC14 GWWG: Talking Points Activity – cultivating exploratory talk through a growth mindset activity

This activity is the one I am most excited about bringing to #TMC14 and to the Group Work Working Group. My intention is to blog more about how this goes during the morning sessions. I also hope that participants will blog more about this too and contribute resources to the wiki.

Exploratory talk is the greatest single predictor of whether group work is effective or not, yet most symmetrical classroom talk (peer talk) is either cumulative (positive but uncritical) or disputational (merely trading uncritical disagreements back and forth).

This activity is based on Lyn Dawes’ Talking Points activity but has been adapted for use within a restorative practices framework. It’s a great way to practice circle skills (i.e., respecting the talking piece) and get students to practice NO COMMENT (i.e., trying to score social points rather than focusing on the task at hand).


  • to support students’ exploratory talk skills by pushing them out of cumulative and disputational modes and into a more exploratory talk mode (i.e., speaking, listening, justifying with NO COMMENT)
  • to reveal student thinking about speaking, listening, justifying and about having a growth mindset 
  • to cultivate a growth mindset community

17 minutes

Get students into groups of three.

Talking Points is a timed activity. Groups will have exactly ten minutes to do as many rounds as they can do. For the whole-class debrief at the end, try to keep track of who thought what and why.

Like classroom circles, Talking Points proceeds in rounds. Each “talking point” statement on the list receives three rounds of attention. One person reads the first statement aloud with NO COMMENT. There are then three rounds of speaking and listening. You want these to be “lightning rounds” rather than plodding or deliberate rounds. The Talking Point statements are provocative and designed to stimulate reactions that can be worked with.

10 minutes

ROUND 1 – Go around the group, with each person saying in turn whether they AGREE, DISAGREE, or are UNSURE about the statement AND WHY. Even if you are unsure, you must state a reason WHY you are unsure. NO COMMENT. You’ll be free to change your mind during your turn in the next round.

ROUND 2 – Go around the group, with each person saying whether they AGREE, DISAGREE, or are UNSURE about their own original statement OR about someone else’s statement they just heard AND SAY WHY. NO COMMENT. You are free to change your mind during your turn in the next round.

ROUND 3 – Take a tally of AGREE / DISAGREE / UNSURE and make notes on your sheet. NO COMMENT

Groups should then move on to the next Talking Point.

At the end of ten minutes ring the bell and have groups finish that round. Don’t let the process go on — they need to stop and move on.

2 minutes

At the end of the ten minutes, give students exactly two minutes to fill out the Group Self-Assessment. They will use this during the whole-class debrief and will hand this in for a group grade.

5 minutes

Ask each group to report out about specifics, such as:

Who in your group asked a helpful question and what was it?
Who in your group changed their mind about a Talking Point? How did that occur?
Who in your group encouraged someone else? How did that benefit the conversation?
Who in your group provided an interesting additional idea and what what is?
Who did your group disagree about and why?

I generally choose two or three of these questions and then move on to the actual mathematical group work while the energy level is still high.

Here are links to the first two documents for this activity:

Talking Points handout 1 – talking about talking
Talking Points – Group Self-Assessment

I'll post links to a whole barrelful of the research as soon as I can!

Sunday, July 20, 2014

TMC #14 Group Work Working Group Morning Session – The Game Plan

There is so much that we could focus on that I felt a strong need to narrow things down. TMC morning sessions seem to work best when they are more “master class” than “introduction,” so I am going to keep us aimed in that direction.

So this three-day morning session will NOT be an introduction to using group work. Rather, we will use the three days as an “Advanced Topics in Improving Group Work” session.

I’ll kick things off with a brief overview presentation on the background, research, and framework for these two topics. Then, we will spend the three days we are going to focus our work in two key areas:
We will explore the need to cultivate a culture of ‘exploratory talk’ in our classrooms (as opposed to ‘cumulative’ or ‘disputational’ talk) and practice ways to develop and deploy exploratory talk tasks. Questions we’ll investigate include — What is ‘exploratory talk’? Who has studied it? Why is it important? How does it improve learning? How is it measured? How does it fit into the learning cycle? What is the evidence-based work that’s been done in this area? How can I harness this work in my classroom?  
We will work together on getting better at improvising group task development and deployment Questions we’ll investigate include — What are the major types of group-worthy tasks? What fits where in the learning cycle? Which types are effective for which processes? How can I develop fluency in developing and deploying the different kinds of tasks in my course area and classroom?
We’ll try out different tasks and task types, analyze them, and split up into sub-groups to sketch out new tasks of the various types that fit what we ourselves will be teaching. Then the whole group can act as a “beta test site” for the tasks we are sketching out.

Each day, we’ll begin with some exploratory talk tasks and structures I’ve researched and will share on the web.

I think it would be amazing if we could also come up with a way of categorizing/labeling different types of tasks to give ourselves more helpful information about what tasks are useful for what purposes.

Looking forward to working with you!

Thursday, July 17, 2014

DIY Geometry Vocabulary Game, courtesy of the MTBoS (a collaborative effort)

Through an amazing collaborative effort on Twitter that took, like, all of 15 minutes, the collective hivemind of the #MTBoS came up with a great way to teach/reinforce vocabulary using Maria Anderson's tic-tac-toe style of Block games.

Most of the needed resources are referenced here on my Words into Math blog post.

I just added a bunch of new files to the Math Teacher wiki, including blank vocab cards so the kids can make up their own practice cards.  I make cards in Pages, so I'm also including the PDF and an exported Word doc version.

BLANK TEMPLATE Words into Math Block game cards.pages

BLANK TEMPLATE Words into Math Block game cards.pdf

1-3 Words into Math Block game cards LEVEL 1 SIDE A.doc

1-3 Words into Math Block game cards LEVEL 1 SIDE B.doc

1-3 Words into Math Block game cards LEVEL 1 SIDE B.pages

1-3 Words into Math Block game cards LEVEL 1 SIDE A.pages

Here's a link to the gameboard.

And voilĂ ! A vocab activity for Geometry is born.

When we receive the Oscar for this production, credit should go to Teresa Ryan (@geometrywiz), Julie Reulbach (@jreulbach), Kate Nowak (@k8nowak , aka The High Priestess), Sam Shah (@samjshah), Tina Cardone (@crstn85), Michael Pershan (@mpershan), and if there's room left in the credits, me too.

UPDATE (08-Aug-14):

Two things:

Thing 1 - I made an actual set of Words into Geometry cards (double-sided) because I want to be ready to have students actively practice. You can find the file here on the Math Teacher's Wiki: Words into Geometry game cards.

Thing 2 - I'm decided to use lima beans and pinto beans as counters for block games in Geometry because I am going to have a lot more students than I've had in the past! These will never go out of style and will always be replaceable as needed.

Monday, July 14, 2014

Life on the Unit Circle - Board Game for Trig Functions

I suppose it was predictable that I would turn the unit circle into a board game as a practice activity, but somehow this still caught me by surprise last night as I was drifting off to sleep. Thank you, deeper wisdom of the unconscious.  :)

This uses the same basic idea as Life on the Number Line, but you can use whatever trig or inverse trig or other practice cards or problems you want to use. Choose your game piece, take turns, and everybody solves whatever problem comes up. Rolling a positive-positive-number or a negative-negative-number on the dice moves your game piece in the counter-clockwise direction (positive angle in standard position) from (0, 1) to start; rolling a positive-negative-number combination moves you in the direction of a negative angle in standard position (clockwise around the circle from wherever you are).

I like this as a practice structure because I try to avoid games that have a strong win-lose association. It also communicates my group work norms of "same problem, same time" and "work in the middle."

You can use the game cards from Trig War (on Sam's blog) or Inverse Trig War (on Jonathan's blog) or from anyone else. I know I plan to. Have kids shuffle them into two piles, one for positive moves and one for negative moves. The point is to do a lot of problems and to get kids used to living on the unit circle. Print the problem cards on colored paper for variety.

I have one new rule, though: whatever point you land on, you have to declare the cosine and sine of that function.

I "score" this as a tournament, requiring each group to do a minimum number of problems, usually a pretty huge number. I keep group scores on a poster or on the white board. I also offer extra credit points for groups that exceed an even more outrageous number of problems. For some reason, the words "extra credit" seem to add a magical flair. I say, who cares as long as kids practice — they're only points!

I buy my plus-minus dice from this seller on eBay and have never had a problem.